An Improved FPT Algorithm for the Flip Distance Problem

TL;DR

This paper presents an improved fixed-parameter tractable (FPT) algorithm for the flip distance problem in triangulations, reducing the number of choices per step from 14 to 5 and improving the overall runtime bound.

Contribution

The authors develop a more efficient FPT algorithm for the flip distance problem by leveraging backtracking and graph analysis, decreasing the complexity of each step.

Findings

Reduced the number of choices per step from 14 to 5.

Established a bound of 2|G| on the action sequence length.

Achieved an improved runtime of O*(k·32^k).

Abstract

Given a set $\cal P$ of points in the Euclidean plane and two triangulations of $\cal P$, the flip distance between these two triangulations is the minimum number of flips required to transform one triangulation into the other. Parameterized Flip Distance problem is to decide if the flip distance between two given triangulations is equal to a given integer $k$. The previous best FPT algorithm runs in time $O^{*}(k\cdot c^{k})$ ($c\leq 2\times 14^{11}$), where each step has fourteen possible choices, and the length of the action sequence is bounded by $11k$. By applying the backtracking strategy and analyzing the underlying property of the flip sequence, each step of our algorithm has only five possible choices. Based on an auxiliary graph $G$, we prove that the length of the action sequence for our algorithm is bounded by $2|G|$. As a result, we present an FPT algorithm running in time $O^{*}(k\cdot 32^{k})$.